# Variational imaginary time evolution: confused with derivation

I am reading the following article, https://arxiv.org/abs/1812.08767, and I am trying to go though the derivation of McLachlan's variational principle for imaginary time evolution. The derivation is done in section B.1.2.

I am confused about going from this equation:

to this equation (assuming real $$\theta_i$$):

I think the $$x$$ at the end is a typo. What I dont understand is how the variation $$\delta$$ removes the need to sum over $$i$$, going from eq. (120) to (123). They also remove the $$\dot{\theta}_i$$ factor and seem to replace it with $$\delta \theta_i$$. I also dont understand why the final term in eq. (120) disappears.

I think I am overall a little confused as to what the variation $$\delta$$ actually does to an equation. Could anyone explain?

• I just learned that that the L2 norm (120) is differentiated with respect to $\dot{\theta}_i$. Nov 15 '20 at 14:53
• This leaves one final confusion. How can one see that the normalization of the wavefunction introduces the $E_\tau$ term in the minimization of the L2 norm? Nov 15 '20 at 14:58